Preheat a grill or grill pan to medium high.
For the sauce: Put the olive oil, butter, anchovy paste, garlic and olives in a small skillet. Warm over medium heat until the butter is melted and the sauce is gently simmering, about 5 minutes. Remove from the heat and stir in the tomatoes and parsley.
For the fish: Sprinkle the halibut fillets on both sides with the salt and drizzle with the olive oil. Grill until grill-marked and opaque throughout, about 4 minutes per side.
Transfer the fish to a serving platter and spoon the bagna cauda over the top.
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